Batshuayi is currently on loan from the Blues to Valencia, though the Liga outfit have made it clear that season-long deal is set to be cancelled as they look to remove his £90,000-per-week salary off their wage bill.
"The player knows,” Valencia general director Mateu Alemany told reporters on Thursday. “For the good of the player and the club, we are looking for a solution.
“We are expecting an exit in the few hours or days.”
Batshuayi has struggled to make an impact for Los Che, scoring just three goals in 23 appearances across all competitions.
He was substituted at half-time during Tuesday's 2-1 loss to Segunda Division outfit Sporting Gijon in the Copa del Rey, with manager Marcelino suggest post-match that "patience had run out" with some of his attacking options.
As such the Belgium international looks set to find pastures new, and the Toffees are hopeful of completing a deal before the end of the January transfer window.
Any move would have to be permanent, given Kurt Zouma is already on loan at Goodison Park and Premier League rules stipulate only player can be loaned to another at any one time in a season.
Chelsea, however, would be resitant to selling Batshuayi for anything less than the £33m ($42m) they paid to sign him from Marseille in the summer of 2016.
Either way, Chelsea are keen to move Batshuayi on despite Maurizio Sarri's side struggling in front of goal in recent weeks.
The Blues have found the net on just five occasions in their last six matches in all competitions, with Sarri keen to add more firepower to his squad in January.
AC Milan's Gonzalo Higuain continues to be linked despite the Chelsea hierarchy believing the deal will be too complicated to complete while Bournemouth's top goalscorer Callum Wilson is also on their radar.
To sign a player Sarri may have to allow Alvaro Morata to leave, however, with the Spain international having held talks with Atletico Madrid and Sevilla in recent days regarding a move away from Stamford Bridge.
Everton, too, have made signing a striker their main priority for the window as Marco Silva looks to lift his side following a festive period which saw them lose three of their four Premier League matches.
Cenk Tosun, who arrived from Besiktas for £27m ($34m) last January, has been on the fringes for much of the campaign, with Richarlison and Dominic Calvert-Lewin prefered as central strikers.
Silva, though, wants to bring in a more consistent goalscoring threat, with Batshuayi seeming to fit the bill despite his career going downhill somewhat after injury brought a premature end to his productive loan spell at Borussia Dortmund during the second half of the 2017-18 campaign.
The potential arrival of Batshuayi could pave the way for Ademola Lookman to leave Goodison Park, with it understood that Southampton boss Ralph Hasenhuttl is keen to reunite with the winger having worked with him at RB Leipzig.